hw6 - determine the most e³cient way to combine these legs...

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AMS/MBA 546 (Fall, 2010) Estie Arkin Network Flows Homework Set # 6 Due Tuesday, October 26, 2010. Suggested reading Chapter 6 of the text [AMO]. 1. Suppose that there are n men, m women and k marriage brokers. Each broker has a list of some of the men and women as clients and can arrange marriages between any pairs of men and women on that list. (Note that each person can be on lists of several brokers.) For this problem, all marriages must be heterosexual and all people are monogomus. (a). Translate the problem of ±nding the most marriages to that of ±nding a max ²ow. Make sure to clearly describe your graph. If you draw the graph, make sure to indicate the source and terminal and all capacities. (b). Assume an additional restriction, that each broker i can arrange at most a i marriages. Modify your formulation from part (a) to ±nd the most marriages. 2. [6.32] An airline has p ²ight legs that it wishes to service by the fewest possible planes. To do so, it must
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Unformatted text preview: determine the most e³cient way to combine these legs into ²ight schedules. The starting time for ²ight i is a i and the ±nishing time is b i . The plane requires r ij hours to return from the destination of ²ight leg i to the start (origin) point of ²ight leg j . Suggest a way to solve this problem using ²ow. 3. [6.40] Given a directed graph G = ( N, A ) with upper bounds on the ²ow u ij which are integer, and nodes s, t ∈ N , we know that there is a max ²ow from s to t that is integer. Suppose that instead, a “helpful” friend gives us a max ²ow x ′ ij where some of the ²ows are noninteger. Show how to convert this ²ow into an integer maximum ²ow. The running time of your algorithm should be O ( mf ) wher m is the number of arcs and f is the number of arcs with fractional (noninteger) ²ow on them. Hint: Send ²ows along cycles....
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This note was uploaded on 01/31/2011 for the course AMS 546 taught by Professor Arkin,e during the Fall '08 term at SUNY Stony Brook.

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